2.7.2 Polynomial Division

Polynomial Division

Polynomial division is the process of dividing two polynomials
- This is usually only useful when the degree of the denominator is less than or equal to the degree of the numerator
To do this we use an algorithm similar to that used for division of integers
To divide the polynomial $P (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + . . . + a_{1} x + a_{0}$ by the polynomial $D (x) = b_{k} x^{k} + b_{k - 1} x^{k - 1} + . . . + b_{1} x + b_{0}$ where k ≤ n
- STEP 1
  Divide the leading term of the polynomial P(x) by the leading term of the divisor D(x) : $\frac{a_{n} x^{n}}{b_{b} x^{k}} = q_{m} x^{m}$
- STEP 2
  Multiply the divisor by this term: $D (x) \times q_{m} x^{m}$
- STEP 3
  Subtract this from the original polynomial P(x) to cancel out the leading term: $R (x) = P (x) - D (x) \times q_{m} x^{m}$
- Repeat steps 1 – 3 using the new polynomial R(x) in place of P(x) until the subtraction results in an expression for R(x) with degree less than the divisor
  - The quotient Q(x) is the sum of the terms you multiplied the divisor by: $Q (x) = q_{m} x^{m} + q_{m - 1} x^{m - 1} + . . . + q_{1} x + q_{0}$
  - The remainder R(x) is the polynomial after the final subtraction

$\frac{P (x)}{a x^{2} + b x + c} = Q (x) + \frac{e x + f}{a x^{2} + b x + c}$ where

The remainder will be linear (degree 1) if e ≠ 0, and constant (degree 0) if e = 0
Note that $P (x) = Q (x) \times (a x^{2} + b x + c) + e x + f$

Synthetic division is a faster and shorter way of setting out a division when dividing by a linear term of the form

To divide $P (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + . . . + a_{1} x + a_{0}$ by $(x - c)$ :

Set $b_{n} = a_{n}$
Calculate $b_{n - 1} = a_{n - 1} + c \times b_{n}$
Continue this iterative process $b_{i - 1} = a_{i - 1} + c \times a_{i}$
The quotient is $Q (x) = b_{n} x^{n - 1} + b_{n - 1} x^{n - 2} + . . . + b_{2} x + b_{1}$ and the remainder is $r = b_{0}$

Given a polynomial $P (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + . . . + a_{1} x + a_{0}$
And a divisor $D (x) = d_{k} x^{k} + d_{k - 1} x^{k - 1} + . . . + d_{1} x + d_{0}$
Write $Q (x) = q_{n - k} x^{n - k} + . . . + q_{1} x + q_{0}$ and $R (x) = r_{k - 1} x^{k - 1} + . . . + r_{1} x + r_{0}$
Write $P (x) = Q (x) D (x) + R (x)$

In an exam you can use whichever method to divide polynomials - just make sure your method is written clearly so that if you make a mistake you can still get a mark for your method!

Perform the division

\frac{x^{4} + 11 x^{2} - 1}{x + 3}

. Hence write

x^{4} + 11 x^{2} - 1

in the form

Q (x) \times (x + 3) + R

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Find the quotient and remainder for

\frac{x^{4} + 4 x^{3} - x + 1}{x^{2} - 2 x}

. Hence write

x^{4} + 4 x^{3} - x + 1

in the form

Q (x) \times (x^{2} - 2 x) + R (x)

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